Transcript Ch 10

Chapter 10:
Analysis of Statically Indeterminate Structures by the Force Method
CIVL3310 STRUCTURAL ANALYSIS
Professor CC Chang
Determinate or Indeterminate ?
Yes
Determinate
Structure
Equilibrium
No
Indeterminate
Why Indetermiante ?
• Advantages
Smaller stresses and deflections
Why Indetermiante ?
• Advantages
Fail safe
1995
Oklahoma City bombing
Prices
• Disadvantages
Stresses due to support settlement
Prices
• Disadvantages
Stresses due to temperature changes
Indeterminate Structures
Symmetric Structures
Axis of symmetry
Structure
Reflection
Identical in geometry, supports and material properties
Symmetrical Structures
Symmetrical Structures
Symmetrical Loadings
Symmetrical Loadings
Anti-Symmetrical Loadings
Anti-Symmetrical
Anti-Symmetrical Loadings
Decomposition of Loadings
(A)
(B) = (A)/2
sum
(C) = Reflection of (B)
(B)+(C)
Symmetrical
(B)-(C)
Anti-symmetrical
Decomposition of Loadings
• Loadings = Symmetrical + Anti-symmetric Loads
=
+
Decomposition of Loadings
Decomposition of Loadings
Analysis of Symmetrical Structures
Loading
Symmetrical
Loading
Anti-symmetrical
Loading
Symmetrical
Structure
Response 1
+
Response
Response 2
Symmetrical Structures under Symmetrical Loads
P
a
P
a
L
Moment & vertical displacement ≠ 0
Slope & axial displacement = 0
P
slope = 0
M≠0
V≠0
Symmetrical Structures under Symmetrical Loads
Symmetrical Structures under Anti-symmetrical Loads
Slope ≠ 0
Moment & vertical displacement = 0
P
a
L/2
a
L/2
P
P
Slope ≠ 0
M=0
V=0
Symmetrical Structures under Anti-symmetrical Loads
Analysis
Analysis
6 degrees of indeterminacy
4 degrees of indeterminacy
4 degrees of indeterminacy
Analysis
Analysis of Statically Indeterminate Structures
• Force methods
 This chapter
• Displacement methods
 Next two chapters
Compatibility
B=0
By
 B  ' B  0
 B  By  f BB  0
Compatibility
A0  MA  fAA  0
Compatibility
B0  By  fBB  0
Compatibility
 D0
D0  Dx  fDD  0
fDD
Compatibility
C C
P
D P
D
 AD
=
A
B
+
f AD
 AD  f AD  FAD  0
1
1
 FAD
Compatibility
DX 0  f DX ,DX  DX  f DX ,DY  DY  0
DY 0  f DY ,DX  DX  f DY ,DY  DY  0
Compatibility
• Settlement
BO  f BB  By  f BC  CY  B
CO  f CB  By  f CC  CY  C
Compatibility
• Settlement
Least Work Method
• Castigliano’s theory
P
?
P
F
M(P,F)
2
M
U(P,F)  
dx
2EI
U
 M  M

 
 EI dx
F F 0  F 

Least Work Method
P
F
M(P,F)
F
2
M
U(P,F)  
dx
2EI
U  M  M
F 
 
 EI dx  0
F  F 
U
0
F
Obtain F
The magnitude of redundant force must be such that
the strain energy stored in the structure is a minimum
Least Work Method
• Virtual work principle
P P
P
UP, F
W  U
W  P   P
M(P,F)
U 
U
U
P   P 
 P 
 F
P
F
U

 U




P

 F  0

P
F
 P

F F
U
U
 P 
 F
P
F
U
 P
P
U
0
F
Castigliano’s theorem
Least work principle
Note: F does not do any work !
Least Work Method
P2
P1
F1

F2
Pm

Fn
Strain energy
UP1, P2 ,, Pm , F1, F2 ,, Fn 
 U
 P   Pi
i
 U

0
 Fi
Forces that do work
Forces that do not do work